Card-counting like a blackjack shark, but in Gloomhaven

Counting cards is the frowned-upon practice of, well, counting cards in games of chance. Card-counting is famous mostly in the context of Blackjack, and is banned by casinos pretty much everywhere.

Luckily, I don’t play Gloomhaven in a casino.


Why count cards?

Obviously, so you can experience what is best in life (e.g. crushing your enemies, seeing them driven before you, and hearing the lamentations of their women).

conan

Seriously. You’re playing this game to the best of your ability, and within the rules. If cards are visible in your discard pile, then by all means ***use that information***!

Your party needs you. Gloomhaven needs you. It’s your duty to use every tool at your disposal, including some simple math.


Dynamic Play

In life, those who can adapt have the best chance to survive. Same goes for board games.

adaptsurvive.jpg

Gloomhaven is not static. The situation changes rapidly, even within one round or turn. So, although we often analyze our cards and abilities from outside the game, testing averages over infinitely many turns, we really should look at the game in the moment. This is the only way to assess current odds and adapt to situations.

 In the shifting sands of playing Gloomhaven, the best tactic is to walk without rhythm.

-Me

Here’s an example: Your stalwart Cragheart has attacked many foes in this scenario, and you noticed that his luck has been great. But, alas – he’s already drawn all the good cards from his attack modifier deck, and was recently Cursed by a strong enemy with a shield. What should he do?

Option A: Play a damaging attack to try to get through the enemy shield, hoping for a “x2” card to be drawn

Option B: Play an attack that has an effect, like stun, immobilize, curse, or disarm

Obviously, the math favors option B. With option A, you would be fishing for just one good card out of many (most good cards are in your discard pile!) hoping to overcome the shield and deal damage. On average, it’s more beneficial here to use an ability that applies an effect, which occurs even if that dreaded “Miss” card is drawn.

We perform this judgement all the time. Why not add a bit of math to it?


Count those cards!

count.jpg

Tactics in Gloomhaven already take into account your discarded attack modifier cards, but usually just as a general “feeling” of what cards are left in that deck. Let’s break down the exact probabilities so we are equipped with hard numbers in our epic quests.

I’ve already discussed the average damage of attack modifier decks, and how these odds change based on alterations to those decks, like added/removed cards from perks. That line of thinking will continue here.

But wait! One major change is that we’re considering instantaneous odds, where before we considered averages over infinitely many draws. One result of that “long-term average” thinking was the odd impact of reshuffled card on average damage. That won’t happen here – we are only asking about the odds of the next card drawn, given the remaining cards in the deck. We don’t care that reshuffling may happen at the end of the turn, since we’re just focusing on this turn!

The odds of missing: Everyone hates drawing that “Miss” card. Let’s analyze how the odds of missing change over time in Gloomhaven for a standard deck (no perk-added cards, no advantage/disadvantage).

There is only one Miss card, and you start out with 20 cards total. The odds of a Miss on your initial draw are simply 1/20 = 5%. Once you have drawn a normal card (anything but Miss or x2 cards that make you reshuffle), your odds of drawing a Miss are now 1 card out of 19 total cards: 1/19 ~= 5.26%. This should be fairly obvious. Here’s a plot of your Miss chance as you keep drawing cards:

chart (9)

From this we see that your Miss odds start at 5%, and keep climbing until there are only 2 cards left (Miss and x2) that both make you reshuffle. Clearly, it is very rare to get that far into you deck before encountering a reshuffle card. On average, a player will draw about 7 cards before having to reshuffle this deck. Once you reshuffle, your odds of a Miss go straight back to 5%.

Note that the odds of a Miss are always going to be the exact same as the odds of a x2.


Other odds

The odds of drawing Miss and x2 were very straightforward. But what if you want to know the odds of something more complicated, like “getting any card better than +0”? Well, at the start, those odds will simply be (number of +1, +2, and x2 cards)/20. Generally, the odds can be described as…

Starting odds = (Number of cards that meet your criteria) / (Total number of cards)

But once you have already drawn some cards, your odds will depend on the contents of your discard pile. So, you’ll want to count the cards in you discard pile that meet your criteria. Let’s abbreviate these names, since they are getting long.

N = total number of cars in your entire deck (for standard deck, this is 20)

D = number of cards in the discard pile

n = number of cards in the whole deck that meet your criteria

d = number of cards in discard pile that meet your criteria

With this, the odds of drawing one of your desired cards can be written as:

Odds  = (n-d) / (N-D)

 

 

Sometimes it’s tough to relate to an equation, so here’s an example: We’ve drawn 5 cards so far from the standard deck: +0, -1, +2, +1, +0. We want to know the odds of getting a +1 or better on the next draw. What are the odds?

N = 20      (total number of cards in the standard deck)

D = 5        (we’ve drawn 5 cards so far and discarded them)

n = 7        (The standard deck has five +1 cards, one +2 card, and one “x2” card)

d = 2        (The discard pile has one +1 card and one +2 card in it)

Odds of getting +1 or better = (7-2)/(20-5)

These odds are 1/3 ~= 33.3%. Knowing this can inform your decisions when attacking this turn!


Is this practical?

Many people (not me) frown at bringing a calculator to the table. I think it can be fun if you also bring a heavy textbook and role-play as a “Hermione Granger” type in your party. It really is pronounced Levi-O-sa, after all.

However, you don’t have to count up specific cards in your discard pile and punch some numbers into a calculator every turn to use this strategy (*But you could!*). If you know your deck’s contents, it’s not hard to have a quick reference sheet handy.

In my next post I’ll show off some references I am making for decks to make this super-easy. I’m also making a small program to show probability distributions, which may be overkill? For now, enjoy taking your Gloomhaven tactics to the next level with a bit of math that is quite simple and impacts your gameplay, but is often forgotten about.

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